Subgroups — C₂×#22549918 #30647315

Order 16256.

536 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861168	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549918	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16861143	C₂	1	2	yes	maximal, complemented
5	C₂×#16861141	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₇	—	2	1	no	—
8	#16861168	—	2	2	no	—
9	#16861168	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₂₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₂₇	—	2	2	no	—
12	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₂₇	—	2	1	no	—
17	C₂×C₄×Dic₁₂₇	—	2	1	no	—
18	#16861143	C₂²	1	8	yes	complemented
19	#16861141	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
22	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
23	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂²×Dic₁₂₇	—	2	4	no	—
25	C₂²×Dic₁₂₇	—	2	2	no	—
26	C₂²×Dic₁₂₇	—	2	1	no	—
27	C₂²×Dic₁₂₇	—	2	1	no	—
28	C₂²×Dic₁₂₇	—	4	4	no	—
29	C₄⋊Dic₁₂₇	—	2	4	no	—
30	C₄⋊Dic₁₂₇	—	2	2	no	—
31	C₄⋊Dic₁₂₇	—	2	1	no	—
32	C₄⋊Dic₁₂₇	—	2	1	no	—
33	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
34	C₄⋊Dic₁₂₇	C₂³	1	4	yes	—
35	C₄⋊Dic₁₂₇	C₂³	1	2	yes	—
36	C₂²⋊Dic₁₂₇	—	2	4	no	—
37	C₄×Dic₁₂₇	—	2	4	no	—
38	C₂⁴×C₁₂₇	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
45	M₁₆×C₁₂₇	C₂×C₄	1	16	yes	complemented
46	C₂×C₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
47	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
48	C₂×Dic₁₂₇	—	2	2	no	—
49	C₂×Dic₁₂₇	—	2	2	no	—
50	C₂×Dic₁₂₇	—	2	4	no	—
51	C₂×Dic₁₂₇	—	4	8	no	—
52	C₂×Dic₁₂₇	—	4	2	no	—
53	C₂×Dic₁₂₇	—	4	4	no	—
54	C₂×Dic₁₂₇	—	4	2	no	—
55	C₂³×C₁₂₇	—	2	2	no	—
56	C₂³×C₁₂₇	—	2	2	no	—
57	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
58	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
59	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
61	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
62	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
67	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
69	C₈×C₁₂₇	C₂²×C₄	1	8	yes	complemented
70	C₂²×C₁₂₇	—	2	2	no	—
71	C₂²×C₁₂₇	—	2	2	no	—
72	C₂²×C₁₂₇	—	2	8	no	—
73	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
74	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
75	C₂²×C₁₂₇	AGL(1,ℤ/8)	1	2	yes	—
76	C₂²×C₁₂₇	#137	1	2	yes	—
77	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
79	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
80	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
81	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
82	Dic₁₂₇	—	4	2	no	—
83	Dic₁₂₇	—	4	4	no	—
84	Dic₁₂₇	—	4	2	no	—
85	C₂×C₁₂₇	—	2	4	no	—
86	C₂×C₁₂₇	#428	1	4	yes	complemented
87	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₁₂₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₁₂₇	C₂×#137	1	1	yes	characteristic
90	C₂×#428	—	127	1	no	maximal, Sylow
91	C₁₂₇	C₂×#428	1	1	yes	characteristic, complemented, Sylow
92	C₂²×M₁₆	D₁₂₇	1	1	yes	characteristic
93	C₂×#117	—	127	1	no	—
94	C₂²×C₄⋊C₄	—	127	1	no	—
95	#428	—	127	8	no	—
96	C₂×#107	—	127	2	no	—
97	C₂×#106	—	127	2	no	—
98	C₂³×C₄	—	254	1	no	—
99	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
100	C₂×M₁₆	Dic₁₂₇	1	8	yes	complemented
101	C₂×M₁₆	C₂×D₁₂₇	1	4	yes	—
102	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
103	#117	—	127	4	no	—
104	#117	—	254	2	no	—
105	C₂×C₄⋊C₄	—	127	2	no	—
106	C₂×C₄⋊C₄	—	127	4	no	—
107	C₂×C₄⋊C₄	—	127	1	no	—
108	C₂×C₄⋊C₄	—	127	1	no	—
109	C₂×C₄⋊C₄	—	254	2	no	—
110	C₂×C₄⋊C₄	—	254	1	no	—
111	C₂×C₂²⋊C₄	—	254	1	no	—
112	C₂×C₄²	—	254	1	no	—
113	#107	—	127	8	no	—
114	#106	—	127	8	no	—
115	C₂⁴	D₅₀₈	1	1	yes	characteristic
116	C₂²×C₄	—	508	4	no	—
117	C₂²×C₄	—	254	4	no	—
118	C₂²×C₄	—	254	1	no	—
119	C₂²×C₄	—	254	1	no	—
120	C₂²×C₄	—	254	2	no	—
121	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
122	C₂²×C₄	D₅₀₈	1	2	yes	—
123	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
124	C₂²×C₄	Dic₂₅₄	1	4	yes	—
125	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
127	M₁₆	C₂×Dic₁₂₇	1	16	yes	complemented
128	C₂×C₈	C₂×Dic₁₂₇	1	8	yes	complemented
129	C₂×C₈	C₂²×D₁₂₇	1	4	yes	—
130	C₄⋊C₄	—	127	2	no	—
131	C₄⋊C₄	—	127	4	no	—
132	C₄⋊C₄	—	127	2	no	—
133	C₄⋊C₄	—	254	4	no	—
134	C₄⋊C₄	—	254	2	no	—
135	C₄⋊C₄	—	254	1	no	—
136	C₄⋊C₄	—	254	1	no	—
137	C₂²⋊C₄	—	254	4	no	—
138	C₄²	—	254	4	no	—
139	C₂³	—	2	2	no	—
140	C₂³	—	2	2	no	—
141	C₂³	C₄⋊Dic₁₂₇	1	4	yes	—
142	C₂³	C₂×D₅₀₈	1	2	yes	—
143	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
144	C₂×C₄	—	508	8	no	—
145	C₂×C₄	—	508	4	no	—
146	C₂×C₄	—	508	2	no	—
147	C₂×C₄	—	508	2	no	—
148	C₂×C₄	—	254	2	no	—
149	C₂×C₄	—	254	2	no	—
150	C₂×C₄	—	254	4	no	—
151	C₂×C₄	C₄⋊Dic₁₂₇	1	8	yes	—
152	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
153	C₂×C₄	C₂²×Dic₁₂₇	1	2	yes	—
154	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
155	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
156	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
157	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
158	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
159	C₂×C₄	C₂³×D₁₂₇	1	1	yes	characteristic
160	C₈	C₂²×Dic₁₂₇	1	8	yes	complemented
161	C₂²	—	2	2	no	—
162	C₂²	—	2	2	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
165	C₂²	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
166	C₂²	#16861237	1	2	yes	—
167	C₂²	#16861242	1	2	yes	—
168	C₂²	C₂²×D₅₀₈	1	1	yes	characteristic
169	C₄	—	508	4	no	—
170	C₄	—	508	2	no	—
171	C₄	—	508	2	no	—
172	C₄	C₂×C₄⋊Dic₁₂₇	1	2	yes	—
173	C₄	C₂×C₄⋊Dic₁₂₇	1	4	yes	—
174	C₄	C₂³×Dic₁₂₇	1	1	yes	characteristic
175	C₄	C₂²×Dic₂₅₄	1	1	yes	characteristic
176	C₂	—	2	4	no	—
177	C₂	#22549918	1	4	yes	direct product factor
178	C₂	C₂²×C₄⋊Dic₁₂₇	1	1	yes	characteristic
179	C₂	C₂×#16861237	1	1	yes	characteristic
180	C₂	C₂×#16861242	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₂₇
Fitting	C₂²×M₁₆×C₁₂₇	C₂
socle	C₂³×C₁₂₇	C₂×D₄
prime radical	C₂⁴×C₁₂₇	D₄
cosocle	C₂×C₄×C₁₂₇	C₂⁴
hypercenter	C₂²×M₁₆	D₁₂₇
hypocenter	C₁₂₇	C₂×#428
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #22549918	C₂	direct (N × H)
4. C₂×#16861143	C₂	semidirect
5. C₂×#16861141	C₂	semidirect
18. #16861143	C₂²	semidirect
19. #16861141	C₂²	semidirect
21. C₂×M₁₆×C₁₂₇	C₄	semidirect
45. M₁₆×C₁₂₇	C₂×C₄	semidirect
46. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
69. C₈×C₁₂₇	C₂²×C₄	semidirect
86. C₂×C₁₂₇	#428	semidirect
91. C₁₂₇	C₂×#428	semidirect
100. C₂×M₁₆	Dic₁₂₇	semidirect
127. M₁₆	C₂×Dic₁₂₇	semidirect
128. C₂×C₈	C₂×Dic₁₂₇	semidirect
160. C₈	C₂²×Dic₁₂₇	semidirect
177. C₂	#22549918	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16861168	2
C₂²×C₄⋊Dic₁₂₇	2
#22549918	2
C₂×#16861143	2
C₂×#16861141	2
C₂²×M₁₆×C₁₂₇	2
C₂×#428	127

Subgroups — C2×#22549918 #30647315

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22549918 #30647315